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A new class of efficient and robust energy stable schemes for gradient flows
Jie Shen, Jie Xu, Jiang Yang
TL;DR
The paper addresses efficient, robust treatment of nonlinear terms in gradient flows while preserving energy stability. It develops SAV schemes with simple linear structure, extends them to challenging applications and higher-order BDF variants, and reports improved efficiency and accuracy, with scope limitations for highly nonlinear dissipation.
Problem
Numerical schemes for gradient flows must preserve energy dissipation while also achieving accuracy, efficiency, and easy implementation.
Method
The paper introduces a scalar auxiliary variable approach that constructs linear, energy-stable schemes using constant-coefficient solves and extends them with higher-order BDF discretizations.
Results
The SAV schemes are reported to be more efficient and accurate than other schemes across a large class of gradient flows and challenging applications.
Takeaways & Limitations
Adaptive SAV schemes provide an efficient approach for a large class of gradient flows, while higher-order BDF combinations improve accuracy in practice.
Takeaways & Limitations
The paper focuses on linear dissipative mechanisms; direct SAV application to highly nonlinear, degenerate, or singular mechanisms may require solving nonlinear equations and may be inefficient.
Abstract
from arXiv · showhide
We propose a new numerical technique to deal with nonlinear terms in gradient flows. By introducing a scalar auxiliary variable (SAV), we construct efficient and robust energy stable schemes for a large class of gradient flows. The SAV approach is not restricted to specific forms of the nonlinear part of the free energy, and only requires to solve {\it decoupled} linear equations with {\it constant coefficients}. We use this technique to deal with several challenging applications which can not be easily handled by existing approaches, and present convincing numerical results to show that our schemes are not only much more efficient and easy to implement, but can also better capture the physical properties in these models. Based on this SAV approach, we can construct unconditionally second-order energy stable schemes; and we can easily construct even third or fourth order BDF schemes, although not unconditionally stable, which are very robust in practice. In particular, when coupled with an adaptive time stepping strategy, the SAV approach can be extremely efficient and accurate.
1. Introduction.
Gradient flows model free-energy-driven dynamics, but energy-stable discretizations must balance dissipation, accuracy, efficiency, and implementation cost. The SAV approach addresses nonlinear terms through broadly applicable schemes designed to retain stability while using simple linear solves.
- Gradient flows are PDE models driven by a free energy and a dissipation mechanism, with applications spanning interfaces, crystallization, thin films, polymers, and liquid crystals.
- Energy dissipation is a central numerical requirement because insufficient consideration can force extremely small time steps to avoid nonphysical results.
- Convex splitting gives unconditional discrete energy decay but generally requires expensive nonlinear solves and lacks a general second-order formulation.
- Stabilization can reduce each step to a constant-coefficient linear solve when the dissipation operator is linear, but a suitable stabilizing operator may not exist and higher-order unconditional stability is generally unavailable.
- IEQ enables linear, second-order, unconditionally energy-stable schemes for broad gradient-flow classes, but typically introduces variable coefficients, coupled multicomponent systems, and lower-bound requirements on the energy density.
- SAV extends IEQ’s advantages while producing constant-coefficient linear equations for single components and decoupled constant-coefficient equations componentwise for multicomponent flows.
- The paper generalizes SAV, applies it to challenging nonlocal and multiphysics models, evaluates accuracy and efficiency, and combines it with higher-order BDF schemes and adaptive time stepping.
2. SAV approach for constructing energy stable schemes.
The SAV approach rewrites a broad class of gradient flows using a scalar auxiliary variable, yielding linear energy-stable schemes that are efficient to implement, including for multiple components.
- SAV formulation: SAV introduces the scalar auxiliary variable r = √E1 for energies whose nonlinear component E1 is bounded from below.A constant can be added to E1 to ensure E1[φ] ≥ C0 > 0 without altering the gradient flow.
- Energy stability: The resulting first-order scheme is unconditionally energy stable with respect to a modified energy.The modified energy can differ from the original energy because r^2 need not equal E1[φ] exactly at discrete times.
- Implementation: Eliminating auxiliary variables reduces the SAV scheme to linear systems that can be solved efficiently using the Sherman–Morrison–Woodbury formula.The implementation requires solving twice a constant-coefficient linear equation of the form (I − ΔtGL)x̄ = b̄.
- Implementation: SAV schemes are extremely efficient when fast solvers are available, whereas convex splitting requires nonlinear systems and IEQ involves variable coefficients.For tensor-product domains with L = −Δ and G = −1 or −Δ, fast solvers can be used.
- Second-order schemes: The SAV/CN scheme is second-order accurate and unconditionally energy stable, while requiring two constant-coefficient linear solves.An explicit approximation of φ(tn+1/2) with O(Δt^2) error is used in the construction.
- Second-order schemes: The SAV/BDF scheme is also second-order accurate and unconditionally energy stable, with its solution obtained through two constant-coefficient linear equations.For multiple gradient-flow components, the corresponding scheme produces decoupled constant-coefficient linear equations after diagonalizing the constant symmetric positive-definite coupling matrix.
3. Numerical validation.
Numerical experiments evaluate SAV schemes on standard, fractional, and nonlocal gradient flows, testing convergence, accuracy, stability, and physical behavior across challenging applications.
- Standard Cahn–Hilliard equation: Second-order convergence is observed for both SAV/CN and SAV/BDF in the standard Cahn–Hilliard equation.The schemes are tested with Fourier Galerkin spatial discretization against an ETDRK4 reference solution.
- Allen–Cahn equation: SAV/CN produces a monotonically decreasing radius close to the sharp-interface limit even with relatively large time steps in the Allen–Cahn benchmark.The computed result is reported as better than a stabilization-method result showing oscillations even at ∆t = 10^-3.
- Allen–Cahn equation: At ∆t = 10^-3, SAV/CN remains close to the reference solution at T = 0.1 and T = 1, whereas IEQ/CN shows visible differences and violates |φ| ≤1.At ∆t = 10^-4, both schemes agree well with the ETDRK4 reference solution.
- Fractional Cahn–Hilliard equation: Smaller fractional order α slows phase separation and coarsening regardless of the mean ¯φ in the fractional Cahn–Hilliard experiment.Configurations are computed at T = 0.032 for different fractional orders and means.
- Phase field crystals: For the nonlocal phase-field-crystal model, both SAV schemes evolve an initial square structure toward a stable hexagonal structure, while SSI remains square and eventually violates energy dissipation.The SSI residual increases for T > 3000 and its free energy eventually increases; both SAV free-energy curves remain dissipative.
4. Higher order SAV schemes and adaptive time stepping.
The paper combines SAV with higher-order BDF schemes and adaptive time stepping to improve accuracy and efficiency for gradient-flow simulations.
- Higher-order SAV schemes: SAV/BDF3 and SAV/BDF4 schemes extend the approach to higher orders, although BDF schemes of order k≥3 are not unconditionally stable.The paper focuses on BDF3 and BDF4 because orders above four do not appear stable.
- Higher-order SAV schemes: BDF4A is unstable, while BDF3A exhibits energy oscillations at Δt = 10^-3; subsequent tests therefore use BDF3B and BDF4B.The BDF3B and BDF4B variants are thereafter denoted BDF3 and BDF4.
- Higher-order SAV schemes: BDF3 and BDF4 achieve third-order and fourth-order convergence rates, respectively.Errors are measured against an ETDRK4 reference solution computed with a sufficiently small time step.
- Higher-order SAV schemes: At Δt = 10^-3, only BDF4 produces the correct solution among BDF2, BDF3, and BDF4; at Δt = 10^-4, all three do.The comparison indicates that higher-order SAV schemes can improve accuracy.
- Adaptive time stepping: Adaptive time stepping is designed to make step size depend on accuracy rather than stability, particularly when solution changes vary across time intervals.The strategy uses a safety coefficient, tolerance, and relative error to update the step size.
- Adaptive time stepping: In the 2D Cahn–Hilliard example, adaptive-time solutions agree well with the small-step reference, whereas large uniform steps do not.The authors conclude that adaptive time stepping is very effective for SAV schemes.
5. Various applications of the SAV approach.
The SAV approach is applied to nonlocal phase-field crystals, molecular beam epitaxy, liquid crystals, and two-phase incompressible flow, producing efficient energy-stable schemes.
- Application scope: The applications include nonlocal phase-field crystals, molecular beam epitaxy without slope selection, Q-tensor liquid crystals, and two-phase incompressible flows.These examples test SAV across gradient flows with different characteristics.
- Nonlocal free energy: For nonlocal free energies, SAV schemes can require only two linear systems and can use common differential operators under an appropriate operator splitting.The scheme is unconditionally energy stable, and choosing L_n1 = 0 and L_n2 = L yields common operators when L > L_n.
- Nonlocal free energy: The nonlocal model is difficult to treat with convex splitting or IEQ approaches, motivating the SAV formulation.The paper explicitly identifies this problem as not easily handled by those approaches.
- Molecular beam epitaxy: For molecular beam epitaxy without slope selection, SAV handles an energy density unbounded from below and yields second-order, linear, unconditionally energy-stable schemes.Each time step requires solving two linear equations of the stated form.
- Liquid crystals: For the Q-tensor liquid-crystal model, the SAV scheme is unconditionally energy stable and can be implemented through coupled second-order equations with constant coefficients.Under periodic boundary conditions, Fourier expansion reduces the computation to independent 5×5 matrix inversions for each mode.
- Two-phase incompressible flow: For two-phase incompressible flow, pressure can be decoupled by projection, while alternative velocity choices trade weak coupling for a decoupled second-order scheme.The cited construction is based on a SAV/BDF2 formulation.
6. Conclusion.
The conclusion presents SAV as an efficient approach for a broad class of gradient flows, while identifying assumptions and open convergence questions.
- Contributions: SAV schemes retain unconditional energy stability, linearity, and second-order accuracy while simplifying implementation through decoupled constant-coefficient linear systems.The scalar auxiliary variable and original unknowns are decoupled at each time step.
- Contributions: SAV requires only the nonlinear integral E1(φ) to be bounded from below and accommodates multiple energy integrals and nonlocal free energies.This broadens the class of gradient flows addressed by the approach.
- Contributions: Higher-order BDF combinations provide stiffly stable schemes, but they are not unconditionally stable.Adaptive time stepping further improves the reported efficiency and applicability.
- Scope and assumptions: The approach requires the linear operator L to contain sufficient dissipative terms, including at least linearized highest-derivative terms.The authors state that E1(φ) must not contain all highest-derivative terms.
- Scope and assumptions: Direct SAV application may be inefficient for highly nonlinear dissipative mechanisms because it can produce degenerate or singular nonlinear equations.The paper gives Wasserstein flows and strongly anisotropic free energies as examples.
- Open questions: Energy dissipation alone does not guarantee convergence, and the paper leaves error analysis and sufficient convergence conditions for future work.The authors plan to establish error estimates for convergence to the original problem.